FDCL Part VIII: Hierarchical Networks and Feedback Control in FDCL Models — Exact Routing, Stability and Observation Conditions
- 1Independent Researcher
Abstract
A finite network reduction is useful only after its input, dynamics and observation are fixed. We study the distinctions it must preserve and the error incurred when one homogeneous reduced kernel approximates a hidden-input process from a fixed initial law. For the native FDCL recurrent readouts, sixteen deterministic states suffice; correlated input requires explicit context. The article extends the observation model to support downsets, sparse input and context-retaining reset, with exact state counts and absorption laws under stated hypotheses. For a specified positive four-letter input, fixed exit proportions yield an exact algebraic all-time marginal total-variation optimum. Freeing the legal pair exits gives the certified enclosure 0.019400280584 <= E*_free < 0.019497701408; the exact free-routing optimum remains open. Coupling nonreturning marked residences allows simultaneous changes to holds and exits and certifies a full sixteen-coordinate neighborhood below 0.0195. Rational flow relaxations, complete covers and polynomial certificates retain explicit finite arithmetic obligations. A complete spatial-and-horizon refinement scheme has a proved lower-value limit. Uniform absorption gives an explicit finite error bound; when every adjustable hold is at most 199/200, the all-time objective equals its prefix through time 2048. These results do not establish termination of the finite adaptive search. The broader manuscript also specifies core-branch routing, weighted diffusion, feedback and commuting dephasing models, with their hypotheses and classical antecedents distinguished from the benchmark-specific certificates. Series and files. FDCL Part VIII, Version 2.0, dated 23 October 2026. This substantially expanded working paper updates version 1.0 at https://doi.org/10.5281/zenodo.23227337. It is not a peer-reviewed publication. The deposit contains the 123-page English manuscript and a clean 196-file archive with complete LaTeX source, six figures, 145 reproducibility files, an evidence guide and a SHA-256 manifest. The bibliography contains 156 entries, including 148 external references. Primary-source comparisons distinguish classical antecedents, their applicable assumptions and the specific FDCL results. Internal editorial and acquisition records and external reference PDFs are excluded. The separate focused version-1.0 code/data record at https://doi.org/10.5281/zenodo.23273215 has a different scope. Verification scope. The package provides file-integrity checks and isolated replay targets. Ten selected verification targets, including the exact fixed-routing proof and the complete free-routing cover with 2781 rational dual certificates, were replayed in preparation; the retained 145 evidence files are unchanged by the subsequent literature and layout revisions. Final package checks and the structural audit pass. Finite certificate acceptance is distinct from the written proofs of all-generation or general assertions. The exact free-routing optimum and minimizing kernel remain unresolved. AI assistance and author responsibility. GPT models and Codex (OpenAI) and Claude models (Anthropic) assisted substantively with mathematical arguments and proof development, verification code, analysis, literature comparison, organization, drafting and revision. Bin Seol formulated the research questions, directed the investigations and assessment, managed research records, and selected and revised the final claims. AI output or agreement between models is not treated as proof. Bin Seol is the sole human author and takes full responsibility for accuracy, integrity and originality. The research received no external funding; no competing interests are declared.
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BibTeX
@article{seol2026fdcl,
title = {{FDCL Part VIII: Hierarchical Networks and Feedback Control in FDCL Models — Exact Routing, Stability and Observation Conditions}},
author = {Bin Seol},
journal = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23227336},
url = {https://doi.org/10.5281/zenodo.23227336},
note = {CC BY},
}
RIS
TY - JOUR
TI - FDCL Part VIII: Hierarchical Networks and Feedback Control in FDCL Models — Exact Routing, Stability and Observation Conditions
AU - Bin Seol
PY - 2026
DA - 2026/10/08
JO - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23227336
UR - https://doi.org/10.5281/zenodo.23227336
AB - A finite network reduction is useful only after its input, dynamics and observation are fixed. We study the distinctions it must preserve and the error incurred when one homogeneous reduced kernel approximates a hidden-input process from a fixed initial law. For the native FDCL recurrent readouts, sixteen deterministic states suffice; correlated input requires explicit context. The article extends the observation model to support downsets, sparse input and context-retaining reset, with exact state counts and absorption laws under stated hypotheses. For a specified positive four-letter input, fixed exit proportions yield an exact algebraic all-time marginal total-variation optimum. Freeing the legal pair exits gives the certified enclosure 0.019400280584 <= E*_free < 0.019497701408; the exact free-routing optimum remains open. Coupling nonreturning marked residences allows simultaneous changes to holds and exits and certifies a full sixteen-coordinate neighborhood below 0.0195. Rational flow relaxations, complete covers and polynomial certificates retain explicit finite arithmetic obligations. A complete spatial-and-horizon refinement scheme has a proved lower-value limit. Uniform absorption gives an explicit finite error bound; when every adjustable hold is at most 199/200, the all-time objective equals its prefix through time 2048. These results do not establish termination of the finite adaptive search. The broader manuscript also specifies core-branch routing, weighted diffusion, feedback and commuting dephasing models, with their hypotheses and classical antecedents distinguished from the benchmark-specific certificates. Series and files. FDCL Part VIII, Version 2.0, dated 23 October 2026. This substantially expanded working paper updates version 1.0 at https://doi.org/10.5281/zenodo.23227337. It is not a peer-reviewed publication. The deposit contains the 123-page English manuscript and a clean 196-file archive with complete LaTeX source, six figures, 145 reproducibility files, an evidence guide and a SHA-256 manifest. The bibliography contains 156 entries, including 148 external references. Primary-source comparisons distinguish classical antecedents, their applicable assumptions and the specific FDCL results. Internal editorial and acquisition records and external reference PDFs are excluded. The separate focused version-1.0 code/data record at https://doi.org/10.5281/zenodo.23273215 has a different scope. Verification scope. The package provides file-integrity checks and isolated replay targets. Ten selected verification targets, including the exact fixed-routing proof and the complete free-routing cover with 2781 rational dual certificates, were replayed in preparation; the retained 145 evidence files are unchanged by the subsequent literature and layout revisions. Final package checks and the structural audit pass. Finite certificate acceptance is distinct from the written proofs of all-generation or general assertions. The exact free-routing optimum and minimizing kernel remain unresolved. AI assistance and author responsibility. GPT models and Codex (OpenAI) and Claude models (Anthropic) assisted substantively with mathematical arguments and proof development, verification code, analysis, literature comparison, organization, drafting and revision. Bin Seol formulated the research questions, directed the investigations and assessment, managed research records, and selected and revised the final claims. AI output or agreement between models is not treated as proof. Bin Seol is the sole human author and takes full responsibility for accuracy, integrity and originality. The research received no external funding; no competing interests are declared.
ER -